## The number of toratopes in n dimensions...

Discussion of shapes with curves and holes in various dimensions.

### The number of toratopes in n dimensions...

...has finally been counted up, not by hand causing a bunch of mistakes, but by a nice mathematical algorithm I worked out, which turns out to be the same as one of Sloane's sequences.

Edit: And then I clicked some links and noticed that this was already done five years ago on this very forum. Well, at least it's properly documented now...!

Keiji

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### Re: The number of toratopes in n dimensions...

Noticing that (pairs of open and closed) toratopes were equivalent to unique series-reduced planted trees inspired me to do some more work on classifying them, and I've discovered some beautiful inter-dimensional symmetry!

I don't have time to make a full explanation right now, but here are all 33 (pairs of) 6D toratopes - each (pair of) toratope(s) is represented by a yellow circle. If you want a nice challenge, try and figure out how my representation works before I have time to explain it

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### Re: The number of toratopes in n dimensions...

I've been getting trees from torotopes too, but mine look different.

In essence, a bracket-notation like ((ii)i) represents a tree. The brackets represent branching points, and the 'i' represent leaves. Here is a little shubbery, to show the latest working diagrams for naming these wretches. The actual section at each level is a sphere, of dimensions as many i's or 'o's that branch out from the previous layer. The ordinary spheres look more like grass, i am afraid.

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`        ((ii)i)      ((ii)(ii))        ((iii)ii)             (iii)   4    i   i         i  i  i  i        | /            \ /   \ /        i  i i   2    o    i          o     o           \|/         \ /             \   /           i o i               i i i   1      o               \ /             \|/                 \|/                           o               o                   o       torus             tiger       bi-spheric comb        sphere`
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### Re: The number of toratopes in n dimensions...

Yes, you're on the way to my trees, I recognise those diagrams from my own work

Take your trees, add an extra 'i' at the bottom, that 'i' you've just added is the yellow node.

Now to save space, represent equivalent nodes as one node with a number in it, and combine equivalent graphs up to yellow nodes. In this representation, toratopic duals e.g. torisphere vs spheritorus appear as different yellow nodes on the same graph! (This has led me to wonder, since there are graphs with more than two yellow nodes, that there may be more to the 'toratopic dual' relation than just two toratopes being 'duals' of each other).

Do the same thing for 6D, and you get the picture I posted above.

Keiji

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### Re: The number of toratopes in n dimensions...

Okay, now I see what the trees represent. Neat! As for the multiple yellow nodes, it may have to do with multiple toratopic duals of the same shape. It seems that there are as many ways to create a dual as there are available radii. Take the 22-torus for example: there are two radii, and reversing them only gives one other dual. However, the ditorus has three, and potentially three ways to switch the radii. Of course, the ditorus has identical n-sphere radii, this would be more effective displayed with a 221-ditorus. Three different radii can allow three different toratopic duals ( trios?), depending on how the 221-ditorus was turned inside out. All three toratopes of 221, 212, and the 311-ditorus are toratopic trios of each other, depending on which radius was switched.
Last edited by ICN5D on Fri Feb 21, 2014 4:04 am, edited 1 time in total.
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### Re: The number of toratopes in n dimensions...

I remember that cool gif animation of the torus turning inside out. It was in reference to the toratopic dual relationship, when I then understood the process. It's a matter of interchanging the dimensions of the radii, like with 22-torus and 31-torus. By setting the lowest dimensional species as the standard minimum, we can derive the free dimensions that can commute. Toratopic duals are related by the number of free dimensions plus available radii.

So, for the 32-torus, we have the 21-torus as the minimum, allowing one free on the major radius and one free on the minor. This makes the 23-torus a dual, as well as the 41-torus. The 41 has 2 free major dimensions, which commute to the minor, making the 23-torus again. So, really the 41, 32, and 23-torii are a toratopic trio.

41 - ((IIII)I)
23 - ((II)III)
32 - ((III)II)

The 41 cannot be turned into a 32 directly, it has to turn into a 23, then a 32. But a 32 can be turned back into a 41! So, really this can be represented by a ring with four nodes, three occupied by the 41, 23, and 32, plus an empty one that shows the break. Then establish a flow with chevrons.

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`                                    ((IIII)I)                                                                                                                        vv^^     ^^                                                                          ((II)III)       ^^                                                                                           vv^^     ^^                                                                                       ((III)II)                                                `

For the 221-ditorus, we have the 211 minimum, allowing 1 free marker plus three radii. However, given the three radii to one free marker ratio, there is no break, all are directly invertible:

221 - (((II)II)I)
212 - (((II)I)II)
311 - (((III)I)I)

Now, with more markers comes more equals, and more radii multiply that effect. Take the 421-ditorus. The 211-ditorus is the standard minimum, allowing 3 free markers and 3 radii open for commuting. Enumerating these will give us 8 equals, the octet of the 511-ditorus:

511 - (((IIIII)I)I)
421 - (((IIII)II)I)
412 - (((IIII)I)II)
322 - (((III)II)II)
232 - (((II)III)II)
223 - (((II)II)III)
241 - (((II)IIII)I)
214 - (((II)I)IIII)

Now, getting to the 223 from the 511 is a complex folding and transforming process, much like a Rubick's Cube. There is a lot of interchanging of free dimensional markers into different radii along the sequence. That would be a cool thing to draw out, the interrelated transforming process tying up all equals into one tree.
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### Re: The number of toratopes in n dimensions...

Nice work, I knew there had to be some important result from the symmetry I found, but I just couldn't put my finger on it

The 41 cannot be turned into a 32 directly, it has to turn into a 23, then a 32. But a 32 can be turned back into a 41!

Hang on, that doesn't make sense - if we are using a process of "break, then do a continuous topological transformation, then join", if you can turn X into Y then you can certainly turn Y into X and vice versa.

Now, with more markers comes more equals, and more radii multiply that effect. Take the 421-ditorus. The 211-ditorus is the standard minimum, allowing 3 free markers and 3 radii open for commuting. Enumerating these will give us 8 equals, the octet of the 511-ditorus:

This doesn't seem right, I'm sure you can't get 8 yellow nodes on a 7D graph, when 6D only goes up to four yellow nodes! Plus, we should be thinking about the expanded rotatope as an invariant. Let me list them as such:

511 - (((IIIII)I)I) - 522 - 522
421 - (((IIII)II)I) - 432 - 432
412 - (((IIII)I)II) - 423 - 432
322 - (((III)II)II) - 333 - 333
232 - (((II)III)II) - 243 - 432
223 - (((II)II)III) - 234 - 432
241 - (((II)IIII)I) - 252 - 522
214 - (((II)I)IIII) - 225 - 522

So as you can see, we have three different expanded rotatopes here. 333, contains only the 322-ditorus. 522, contains three ditoruses: 511, 241 and 214. 432 contains the remaining four: 421, 412, 232 and 223. Which means 333 would be a yellow node in a graph with no others, the graph of 511/241/214 would have three yellow nodes (one for each of them) and the graph of 421/412/232/223 would have four yellow nodes (one for each of them). They could then only be transformed into the other yellow nodes in their graph.

Keiji

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### Re: The number of toratopes in n dimensions...

You know, you're right. I didn't realize the expanded notation was what I should be using. It's going to be a little confusing with using Marek's latest notation for them. But, at least I got the multiple nodes thing down! The 322 is a great example. Some of them have a quartet of toratopic equals, like the 332. If going into tritorii and especially tetratorii the nodes will pile up in a single tree for sure.
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### Re: The number of toratopes in n dimensions...

I was playing around with the toratope notation, to see if there were a fast way to count them, and to see what it might mean.

The toratopes developed by the rotatope notation, one and all, are comb products. The difference between a simple comb and the right comb is the direction of the axis to which they are applied. Even the tiger can be made by a fairly regular process. The comb product is a pondering product: that is, when you multiply two figures, one supposes that that one draws perpendiculars to the surface on the subject, and these become a axial direction of the object.

In the simple torus product, each new element is an n-sphere that is used to replace the surface radial on the previous. If we suppose in circle ## circle ## circle, the last two circles represent a torus, the pondered axis runs along a set of spokes of the wheel. In the tiger, it's the same thing, except that the pondering axis runs with the axle of the wheel. So if the first circle is in the wx plane, and the torus is yz ## ra. When we apply the torus product here, instead of adding it to the major axis, which would give wx ## ry ## rz, we apply the pondering axis to the 'axle', which runs in the direction of 'a', so we get wx ## yz ## rr.

When we look at a toratope tree, every branching point (o in my earlier diagram), becomes a sphere of p dimensions, where there are p points (either o or i), directly connected above it. So one can see directly the surface of say, a tiger, is a tri-circular prism. One can for two different toruses, shuffle the elements of the surface, but can't migrate from say (circle-glome) to (sphere-sphere), even though both of these are 5d.

Another constraint is that it's only a surface. The 'inside' and 'outside' do not have to be topologically equal, even though they are in 3d. So a circle-sphere comb and sphere-circle comb has the same surface, but the insides are different. You can turn one inside out to the other, but you can't transform the solid. You can't turn something like a glome-circle into a circle-circle-circle, or the like.

Because the comb-product really is a 'product of surfaces', we can construct it by using a slightly different form. We remove one of the I's from the inner product, because it represents the first radius.

The simple toruses have all brackets open before the first closes, and must have an outer set. That is, for 6D we could write it as [i.i.i.i.i), where [ opens all brackets, and . is freely nothing or a bracket. This means there are 2^4 of these.

When we come to count the rest, we draw trees, which have weight. The minimum weight of a node on a tree is 1, but if there are x branches connected under it, then the minumum is raised to x-1.

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`     w=1                w=3             w=5           w=5                                                      1                                                     / \                         1              2           1   1                        / \            /|\             / \       1               1   1          1 1 1           1   1   2    1                                                                   1    3D   3    3            1,11   1                                               4    4D   4    7            2,11   2                                              11    5D                     1,21   2   5   15            3,11   4        2,111  2      11111     1             33    6D                     1,31   4                     2,21   4                     1,22   3   6   31            4.11   2*8     3,111  4       211111   4*2            93    7D                     3.21   3*8     2,211  4                     2.22   1*6`

Let's look at how this table is made. The toratopes are taken as a series of nested 'simple' torotopes. So something for (((ii)i)(iii)) we see that there are two inner sets of brackets, and we expand these as far as the simples go, we get then (AB), where A = ((ii)i) and B = (iii). We can then see that C = (AB) is a torus, so we have the first diagram C above A, B.

The nodes must contain a certain number of i's, which is given by the formula above. This gives the base weight of the tree. Anything else can be freely scattered with no heed to the structure. So the first structure has a weight of three, there are two free coins we can scatter over the circles. In 7D, there would be three free coins here, and one for the next two. So while in 6D there is only one tree of the third type, in 7D, the free disk can go on any of the five stations, and the total is 2*1*1*1*1 * 5 = 10 different torii in this tree.

Apart from this,

We count the number of elements, i's or A, B, C, ... in each layer, starting from 0. We get A=B=2, and C=1. So this torus is one of the three in the layer 1,22.

We see that 1-> 1, 2-> 2 , 3->4 . 4-> 8, and 5->16 simple torotopes as before. 2,21 has four because the 2's are separate levels. In 1,22, the two '2's are on the same leaf, and so we have to account for that if A<>B, then 2*2 would count AB and BA, while 2(2+1)/2 would count AB and AA only once. It looks like the trees have an odd weight, so i would be supprised to find a tree of weight 6.

For 5d, we have tree w=1, giving 7, and tree 2, giving 1,21 and 2,11, each of 2, gives all together 11 torii.

For 7d, we would have w1 giving 31, the two large trees give 8 and 10 respectively. The first tree, has

4,11 1,41 3,21 2,31 1,32, each of order 8; ie 40
2,22 gives 6

So there are 31 + 46 + 8 + 10 = 95 torotopes in that dimension.
Last edited by wendy on Sun Feb 23, 2014 6:57 am, edited 1 time in total.
Reason: Fixed number of toratopes in 7D.
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### Re: The number of toratopes in n dimensions...

Got me thinking... tiger, ((II)(II)) is circle x circle where every point of ridge is replaced by a circle. In ((II)(II)I) every point of ridge is replaced by a sphere. If every point of ridge is replaced by a torus, we'll get (((II)(II))I). But what happens if every point of ridge is replaced by a tiger? I suspect that this is not in the toratope notation since it's not clear how the tiger would be oriented...
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### Re: The number of toratopes in n dimensions...

Marek, I think that's the double tiger (((II)(II))(II)), 22020-tiger. If I remember the cuts correctly, one of the 3D's is a quartet of tiger-cuts ( two vert stack torii) in vert of a square. This certainly looks like a tiger along the two ortho circles.
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### Re: The number of toratopes in n dimensions...

ICN5D wrote:Marek, I think that's the double tiger (((II)(II))(II)), 22020-tiger. If I remember the cuts correctly, one of the 3D's is a quartet of tiger-cuts ( two vert stack torii) in vert of a square. This certainly looks like a tiger along the two ortho circles.

But in that case there is a simple rule!

((II)(II)) has circle (II)
((II)(II)I) has sphere (III)
(((II)(II))I) has torus ((II)I)
(((II)(II))(II)) has tiger ((II)(II))
In all cases, the "(II)(II)" string is replaced with "II". This is intriguing.
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### Re: The number of toratopes in n dimensions...

Yes!! I instantly recognized it by your description. I love this forum, I've been learning so many awesome things lately. Shapes are like numbers, or matrices, with intricate combinations unique to each one. I'm going to blow everyone's mind with what I've been working on, when I get home. I found a way to enumerate ALL n-cells of every rotatope. This will allow us to complete the wiki, and even more.
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### Re: The number of toratopes in n dimensions...

ICN5D wrote:Yes!! I instantly recognized it by your description. I love this forum, I've been learning so many awesome things lately. Shapes are like numbers, or matrices, with intricate combinations unique to each one. I'm going to blow everyone's mind with what I've been working on, when I get home. I found a way to enumerate ALL n-cells of every rotatope. This will allow us to complete the wiki, and even more.

You know, this sounds suspiciously like something a movie character says minutes before his mysterious death which leaves the problem in question unresolved until sequel.
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### Re: The number of toratopes in n dimensions...

ICN5D wrote:This will allow us to complete the wiki, and even more.

The wiki can never be completed, since you can never discover all shapes that could possibly exist

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### Re: The number of toratopes in n dimensions...

You are correct At least I can do all existing ones! Infinity is a pretty large number. The sequel will arrive sometime tonight, eastern time, which seems to be morning for most of you.
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### Re: The number of toratopes in n dimensions...

Okay, still working on the n-cell computations. But, now I have a question: earlier in " Tiger Explained", I made the distinction that the torus*torus prism was the exact same as the ditorus*circle prism. So, ((II)I)((II)I) == (((II)I)I)(II) , where both have two ortho bound tritoruses as surtope elements. This makes me wonder about the tigroids of these two: (((II)I)((II)I)) = 21210 is the duotorus tiger, and ((((II)I)I)(II)) = 21120 is the cylditorintigroid. It seems to me that if they are cartesian product equals, then wouldn't their tigroids be equal as well? Even if we're multiplying the hollow forms of both shapes then inflating with a circle, their surfaces are interchangeable, like a toratopic dual, but in a tigroid symmetry. Also take the 3120-torus tiger (((III)I)(II)), it's cartesian dual is the 2130-torus tiger (((II)I)(III)). Maybe we can call them "tigroidal duals" or something to that effect. It also seems like this kind of dual doesn't occur until 5 or 6D, where the cartesian product commuting can happen. Then, of course, this effect can have trios and quartets of tigroidal equals in higher shapes.
Last edited by ICN5D on Sun Feb 23, 2014 1:31 am, edited 2 times in total.
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### Re: The number of toratopes in n dimensions...

One must realise that a new toratope can be made by taking a toratope I = (...), and A = (...), and replacing any i in I, by a copy of A. This means, for example, that every toratope has a presentation that consists of a tree of n-spheres (eg tiger = (ab), A=(ii). B=(ii), gives ((ii)(ii)).

The toratope that is being nested is the larger shape, the frame that the nest (or 'a' is in), is the cross section. So Keiji's question about a tiger-tiger, is simply a tiger-shaped cross-section ((ii)(it)) where the 't' is replaced by a tiger ((ii)(i((ii)(ii)))

The letter we replace in the cross section (t in the above example), is the axis of pondering. That is, the lines of normal that we draw to the surface of the greater figure, becomes an axis of the lesser figure. In some cases, there are more than one axis to look at. Consider a circle as the larger figure, and the torus as the section.
• ((ai)i) A = (ii). Here, the axis of pondering runs as the spokes of a wheel would in ((ai)i) : it's parallel to one of the diameters of the greater circle. The outcome here is a di-torus.
• ((ii)a) A=(ii) Here the axis of pondering runs as the axle of the wheel, ie not in the inner circle (ii). When we add this as a cross section to the circle, we see the axis lost is parallel to the wheel-axle, and the full circles are preserved in prism-product, the lesser diameter of the wheel (ie tube diameter), now serves both circles.
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### Re: The number of toratopes in n dimensions...

I also wanted to throw out an extension to the current toratope notation, that includes bracketopes for crosscuts other than circles. I either need something like this, or I can use my own notation, to describe the curved cells of rotatopes. Some of them are things like a line ditorus, cube torus, cone torisphere, etc. When I enumerate some of these upcoming wild curved n-surfaces, there should be a commonly accepted notation, understood by the masses of HDDB. The radii are reverse in these as in ([minor]major), because the cartesian product can allow it from (major[minor]). The spheration happens in a hose-linking of the prism extension, or height into a circle. Start with [...], extrude into [...]I, then hose-link the prism into a torus ([...]I). The [...] term will always be the crosscut, where [] is a 0D point.

([]I) - glomolatrix, 1D, (O), 1-cell of cylinder, circle

([I]I) - line torus, 2D, |(O), 2-cell of cylinder
([]II) - glomohedrix - (OO), 2-cell of sphere
(([]I)I) - (O)(O), 2-surface of torus, torohedrix
([]I)([]I) - duocylinder margin, [(O)(O)], 2-cell of duocylinder, cylconinder

((II)I) - circle torus - |O(O), 3-cell of duocylinder, torinder
([II]I) - square torus - ||(O), 3-cell of coninder, cubinder
([I>]I) - triangle torus - |>(O), 3-cell of cyltrianglinder, cylindrone, dicone
(([I]I)I) - line ditorus - |(O)(O), 3-cell of cylconinder, torinder
([I]II) - line torisphere - |(OO), 3-cell of spherinder
([]III) - glomochorix - (OOO), 3-cell of glome

([III]I) - cube torus - |||(O), 4-cell of tesserinder, cone diprism
([(II)I]I) - cylinder torus, torinder = ((II)I)I - |O|(O), 4-cell of cylconinder
([(II)>]I) - cone torus - |O>(O), 4-cell of cylconinder
([II>]I) - square pyramid torus - ||>(O), 4-cell of cylhemoctahedrinder
([I>I]I) - triangle prism torus - |>|(O), 4-cell of cyltriandyinder, diconinder, cubindrone, contrianglinder
([I>>]I) - tetrahedron torus - |>>(O), 4-cell of tricone, dicylindrone, cyltetrahedrinder
([II]II) - square torisphere - ||(OO), 4-cell of cubspherinder
([I>]II) - triangle torisphere - |>(OO), 4-cell of sphentrianglinder
((([I]I)I)I) - line tritorus - |(O)(O)(O), 4-surface of ditorinder ( ditorus prism)
([I]III) - line glomitorus - |(OOO), 4-cell of glominder

--Philip

EDIT: I changed the " ' " symbol to the " > " to make it easier to see it, ' is so small, it disappears in the brackets.
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### Re: The number of toratopes in n dimensions...

I've been thinking more about that tigroidal cartesian dual thing. I'm not sure which toratope tree that Keiji made are the 6D tigroids. But, when using expanded toratope notation, the 21210-duotorus tiger and the 21120-cylditorintigroid are both (2222), and 2222 for their open toratope version. Interchanging the hollow circle around from the second torus to the first: ((II)I)((II)I) to (((II)I)I)(II) will switch the 2's around and having no effect in 2222 . Since the tigroid is just an inflated cartesian product of the surfaces of both shapes, this property will still exist within the tiger symmetry. No matter which arrangement is used, both will still have the same 2 tritoruses ((((II)I)I)I) that are ortho bound just like that of the duocylinder. The margin of these is the cartesian product of four hollow circles, derived from the 2x2 or the 3x1 radii when combined.

(II) - circle, 2
surface: 1 point torus ([]I) containing ∞ 0D vertices

(II)(II) - duocylinder, 2*2
Surface: 2 ortho bound torii ((II)I)
Margin: ([]I)([]I), 2 hollow circles, 2D edge containing ∞ 1D edges, ∞ 0D vertices

((II)I)(II) - cyltorinder, 22*2
Surface: 2 ortho bound ditoruses (((II)I)I)
Margin: ([]I)([]I)([]I), 3 hollow circles, 3D edge containing ∞ 2D faces, ∞ 1D edges, ∞ 0D vertices

((II)I)((II)I) - bi-toric prism, 22*22
surface: 2 ortho bound tritoruses ((((II)I)I)I)
margin: ([]I)([]I)([]I)([]I), 4 hollow circles, 4D edge containing ∞ 3D cells, ∞ 2D faces, ∞ 1D edges, ∞ 0D vertices

(((II)I)I)(II) - cylditorinder, 222*2
surface: 2 ortho bound tritoruses ((((II)I)I)I)
margin: ([]I)([]I)([]I)([]I), 4 hollow circles, 4D edge containing ∞ 3D cells, ∞ 2D faces, ∞ 1D edges, ∞ 0D vertices

((II)I)I)((II)I) - ditorus*torus, 222*22
surface: 2 ortho bound tetratoruses (((((II)I)I)I)I)
margin: ([]I)([]I)([]I)([]I)([]I), 5 hollow circles, 5D edge containing ∞4D cells, ∞ 3D cells, ∞ 2D faces, ∞ 1D edges, ∞ 0D vertices

(((II)I)I)I)(II) - cyltritorinder, 2222*2
surface: 2 ortho bound tetratoruses (((((II)I)I)I)I)
margin: ([]I)([]I)([]I)([]I)([]I), 5 hollow circles, 5D edge containing ∞4D cells, ∞ 3D cells, ∞ 2D faces, ∞ 1D edges, ∞ 0D vertices

and the general pattern is the commuting " * " within the 2's. If there are more dimensions in the radius than 2, those numbers also can commute as in 32*2 == 22*3
in search of combinatorial objects of finite extent
ICN5D
Pentonian

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### Re: The number of toratopes in n dimensions...

ICN5D wrote:But, when using expanded toratope notation, the 21210-duotorus tiger and the 21120-cylditorintigroid are both (2222), and 2222 for their open toratope version.

Hold up - just want to clear up some terminology here:

*expanded toratope notation does not exist, there is a notation called extended toratope notation which is unrelated to what we are doing here, and there is the expanded rotatope which is a property of toratopes (which is written in rotopic digit notation)
*the expanded rotatope never has brackets in/around it
*the expanded rotatopes of ((II)I)((II)I) and (((II)I)I)(II) (which are open toratopes) are both 2222
*the expanded rotatopes of (((II)I)((II)I)) and ((((II)I)I)(II)) (which are the closed versions) are both 22222

Hope that helps

Keiji

Posts: 1962
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

### Re: The number of toratopes in n dimensions...

Oh, thanks for that I meant extended toratope notation. As for the bracketopes inside the toratopes, it's an experiment. An attempt to have a way to describe those with non-circular cross cuts. But, if you can tell what the ||>(OO) is, then I don't need to create anything new.

Other than that, what do you think about the cartesian dual relationship? More so, the tigroids of those open rotatopes? I think I found a new kind of dual property, in the tiger symmetry. But, no one is jumping to explain it. Has anyone on here ever addressed this before? Does it really mean anything at all?

I put that mind-blowing expansion to my algortihm, where I enumerate all n-cells, here: viewtopic.php?f=3&t=1857#p20573
in search of combinatorial objects of finite extent
ICN5D
Pentonian

Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL